The present invention relates generally to dither for quantizing devices, and more particularly to the correction of quantization errors in electonic devices or systems such as analog-to-digital converters (ADCs), and more specifically multi-stage ADCs.
Fourier theory requires that:                a) all repetitive signals of period T may be expressed as a sum of a DC offset, a fundamental sine wave of period T, and the harmonics of the fundamental;        b) the fundamental and harmonics each have an amplitude and phase that is uniquely defined by the shape of the repetitive signal.A sine frequency is the reciprocal of its period or cycle time, so the fundamental frequency is 1/T cycles per second or Hertz. A harmonic of a sine wave of period T is a frequency that is an integer multiple of the fundamental frequency so the nth harmonic has frequency n/T or a period of T/n.        
In mathematical terms a real signal Y(t) that repeats with frequency f may be expressed as:       Y    ⁡          (      t      )        =            D      ⁢                          ⁢      C        +                  ∑                  n          =                      1            -            M                              ⁢              (                              A            ⁡                          (                              n                ,                f                            )                                ⁢                      sin            ⁡                          (                                                2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                  nft                                +                                  P                  ⁡                                      (                                          n                      ,                      f                                        )                                                              )                                      )            A(n, f) is the amplitude of the nth harmonic for the fundamental frequency and P(n, f) is the corresponding phase. Also 2πf=ω in radians/second, the unit of angular frequency, so there are 2π radians in a circle or cycle. In realizable systems the passage of high frequencies, and therefore high harmonics, is limited. For example analog systems are limited by maximum drive current capabilities and capacitive loading so in practice M does not extend to infinity. A repetitive signal may then be defined in terms of T and two finite sets of M amplitudes {A} and phase {P}.
If the fundamental frequency is applied to the input of a non-linear device such as a quantizing device, then the output consists of the fundamental itself and its various harmonics. As the input has no harmonics, the amplitude of each harmonic of the system output relative to the amplitude of the fundamental is a measure of its harmonic distortion. Spurious Free Dynamic Range (SFDR) is a measure of the relative size of the largest harmonic with respect to the fundamental for a defined range of pure sine-wave input frequencies. ADCs usually have a much better measured SFDR for inputs having more frequency components.
ADC quantization distortion has a certain amount of energy associated with the quantization voltage step. The Fourier Theorem implies that harmonic distortion only lies at frequencies related to those at the input. Inputs that have more frequency components spread the quantization noise energy out into many intermodulation and harmonic distortion components, usually reducing the amplitude of each harmonic component. Conversely harmonic distortion in ADCs is most apparent for single sine-wave inputs. In this case quantization harmonics may be severe, as no intemodulation mechanism is present, so the quantization energy that is not noise manifests itself as harmonic distortion. Furthermore minor changes in the input signal cause large changes in the harmonic profile, so any distortion cancellation mechanism is unable to cancel quantization harmonic distortion. Finally even high harmonics may have considerable energy—limited only by the bandwidth of the ADC.
The “fingerprints” of quantization distortion are:                a) rapid changes in harmonic profiles with even small changes in the input;        b) if the input sinusoid changes smoothly, the harmonics seem to vary relative to each other, but over a similar amplitude range; and        c) the harmonic profile extends out to the maximum bandwidth allowed by the analog system.Dither may be added to ADC inputs to reduce quantization distortion. The effectiveness of dither relies on the fact that even small inputs to an ADC cause quantization harmonic distortion, as the distortion mechanism is a natural consequence of transitioning between voltage levels on a periodic basis. In fact the quantization distortion is closely related to the ADC quantization step sizes being exercised, and not to the size of the input signal.        
Dither in the form of noise may be added into the input of an ADC to help spread the energy of the quantization noise so that it is no longer a problem. Unfortunately some multi-stage ADCs produce quantization distortion at earlier stages in the ADC pipeline, so dither added into the signal may have to be much larger to reduce quantization effects. The large amplitude dither is needed because multi-stage pipelined ADCs require large amplitude to excite dither in the least-significant bit (LSB) of the most significant stage of the ADC in order to reduce the quantization noise. This most significant stage may be only a few bits, so dither around 1/16 of the total input range may be needed. If large amplitude signals are added into the ADC input, they must be accurately subtracted from the ADC digital output.
An approach for adding and canceling a relatively large dither signal is to generate a large, accurate digital sampled sine wave, convert it to an analog signal, add it to the ADC input, and then subtract it out digitally from the ADC output. Such a dither circuit for improving quantization distortion in analog-digital and digital-analog conversion is shown in U.S. Pat. No. 4,812,846 where dither in the form of a frequency signal at one-half of the sampling frequency is added to an input signal and subtracted from an output signal of the converter. The dither signal may be easily generated as it is at ½ fs, so a digital divide-by-two on the clock generates the required frequency with all its harmonics at fs/2 or at DC. The disadvantage is that at each sampling instant the dither has only two different added voltage levels, with limited effectiveness in the spreading of quantization energy.
What is desired is a dither method for a quantizing device, especially a multi-stage analog-to-digital converter (ADC), that reduces quantization noise more effectively.